Proving Injectivity of Linear Functions

Prove that the function $f(x) = 3x - 2$ is injective.

Proving the injectivity of a function generally involves showing that if two inputs give the same output, then these inputs are actually the same. For the function $f(x) = 3x - 2$, which is linear, this boils down to solving the simple equation $f(a) = f(b)$ and showing that this implies $a = b$. This high-level approach revolves around understanding the definition of an injective function, or a one-to-one function, which does not map distinct inputs to the same output.

The relevance of injectivity in mathematics lies in its role in understanding the behavior of functions, particularly in determining whether a function can have an inverse. A function is invertible if and only if it is both injective and surjective. Linear functions like $f(x) = 3x - 2$ are especially straightforward to analyze for injectivity because they inherently map their domain to the codomain in a "straight-line" manner, thus avoiding repeated outputs. Hence, checking such functions for injectivity serves as a foundational exercise in much of higher mathematics, illustrating how basic algebraic techniques are applied in function analysis.

Furthermore, proving the injectivity of functions is an essential skill in fields such as calculus and algebra where understanding the peculiarities of function behavior can lead to more profound insights about continuity, differentiability, and other deeper concepts regarding how functions operate and interact.